Questions D1 (899 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
AQA D1 Q5
5 [Figure 1, printed on the insert, is provided for use in this question.]
The network shows the times, in minutes, to travel between 10 towns.
\includegraphics[max width=\textwidth, alt={}, center]{194d16e0-8e05-45c0-8948-99808440ed2a-006_412_1561_568_233}
  1. Use Dijkstra's algorithm on Figure 1 to find the minimum time to travel from \(A\) to \(J\).
    (6 marks)
  2. State the corresponding route.
    (1 mark)
AQA D1 Q7
7 Stella is visiting Tijuana on a day trip. The diagram shows the lengths, in metres, of the roads near the bus station.
\includegraphics[max width=\textwidth, alt={}, center]{194d16e0-8e05-45c0-8948-99808440ed2a-007_1539_1162_495_424} Stella leaves the bus station at \(A\). She decides to walk along all of the roads at least once before returning to \(A\).
  1. Explain why it is not possible to start from \(A\), travel along each road only once and return to \(A\).
  2. Find the length of an optimal 'Chinese postman' route around the network, starting and finishing at \(A\).
  3. At each of the 9 places \(B , C , \ldots , J\), there is a statue. Find the number of times that Stella will pass a statue if she follows her optimal route.
AQA D1 2006 January Q1
1
  1. Draw a bipartite graph representing the following adjacency matrix.
    (2 marks)
    \(\boldsymbol { U }\)\(V\)\(\boldsymbol { W }\)\(\boldsymbol { X }\)\(\boldsymbol { Y }\)\(\boldsymbol { Z }\)
    \(\boldsymbol { A }\)101010
    \(\boldsymbol { B }\)010100
    \(\boldsymbol { C }\)010001
    \(\boldsymbol { D }\)000100
    \(\boldsymbol { E }\)001011
    \(\boldsymbol { F }\)000110
  2. Given that initially \(A\) is matched to \(W , B\) is matched to \(X , C\) is matched to \(V\), and \(E\) is matched to \(Y\), use the alternating path algorithm, from this initial matching, to find a complete matching. List your complete matching.
AQA D1 2006 January Q2
2 Use the quicksort algorithm to rearrange the following numbers into ascending order. Indicate clearly the pivots that you use. $$\begin{array} { l l l l l l l l } 18 & 23 & 12 & 7 & 26 & 19 & 16 & 24 \end{array}$$
AQA D1 2006 January Q3
3
    1. State the number of edges in a minimum spanning tree of a network with 10 vertices.
    2. State the number of edges in a minimum spanning tree of a network with \(n\) vertices.
  2. The following network has 10 vertices: \(A , B , \ldots , J\). The numbers on each edge represent the distances, in miles, between pairs of vertices.
    \includegraphics[max width=\textwidth, alt={}, center]{4a186c87-5f84-4ec3-8cc3-a0ed8721b040-03_1294_1118_785_445}
    1. Use Kruskal's algorithm to find the minimum spanning tree for the network.
    2. State the length of your spanning tree.
    3. Draw your spanning tree.
AQA D1 2006 January Q4
4 The diagram shows the feasible region of a linear programming problem.
\includegraphics[max width=\textwidth, alt={}, center]{4a186c87-5f84-4ec3-8cc3-a0ed8721b040-04_1349_1395_408_294}
  1. On the feasible region, find:
    1. the maximum value of \(2 x + 3 y\);
    2. the maximum value of \(3 x + 2 y\);
    3. the minimum value of \(- 2 x + y\).
  2. Find the 5 inequalities that define the feasible region.
AQA D1 2006 January Q5
5 [Figure 1, printed on the insert, is provided for use in this question.]
The network shows the times, in minutes, to travel between 10 towns.
\includegraphics[max width=\textwidth, alt={}, center]{4a186c87-5f84-4ec3-8cc3-a0ed8721b040-05_412_1561_568_233}
  1. Use Dijkstra's algorithm on Figure 1 to find the minimum time to travel from \(A\) to \(J\).
    (6 marks)
  2. State the corresponding route.
    (1 mark)
AQA D1 2006 January Q6
6 Two algorithms are shown. \section*{Algorithm 1}
Line 10Input \(P\)
Line 20Input \(R\)
Line 30Input \(T\)
Line 40Let \(I = ( P * R * T ) / 100\)
Line 50Let \(A = P + I\)
Line 60Let \(M = A / ( 12 * T )\)
Line 70Print \(M\)
Line 80Stop
\section*{Algorithm 2}
Line 10Input \(P\)
Line 20Input \(R\)
Line 30Input \(T\)
Line 40Let \(A = P\)
Line 50\(K = 0\)
Line 60Let \(K = K + 1\)
Line 70Let \(I = ( A * R ) / 100\)
Line 80Let \(A = A + I\)
Line 90If \(K < T\) then goto Line 60
Line 100Let \(M = A / ( 12 * T )\)
Line 110Print \(M\)
Line 120Stop
In the case where the input values are \(P = 400 , R = 5\) and \(T = 3\) :
  1. trace Algorithm 1;
  2. trace Algorithm 2.
AQA D1 2006 January Q7
7 Stella is visiting Tijuana on a day trip. The diagram shows the lengths, in metres, of the roads near the bus station.
\includegraphics[max width=\textwidth, alt={}, center]{4a186c87-5f84-4ec3-8cc3-a0ed8721b040-06_1539_1162_495_424} Stella leaves the bus station at \(A\). She decides to walk along all of the roads at least once before returning to \(A\).
  1. Explain why it is not possible to start from \(A\), travel along each road only once and return to \(A\).
  2. Find the length of an optimal 'Chinese postman' route around the network, starting and finishing at \(A\).
  3. At each of the 9 places \(B , C , \ldots , J\), there is a statue. Find the number of times that Stella will pass a statue if she follows her optimal route.
AQA D1 2006 January Q8
8 Salvadore is visiting six famous places in Barcelona: La Pedrera \(( L )\), Nou Camp \(( N )\), Olympic Village \(( O )\), Park Guell \(( P )\), Ramblas \(( R )\) and Sagrada Familia \(( S )\). Owing to the traffic system the time taken to travel between two places may vary according to the direction of travel. The table shows the times, in minutes, that it will take to travel between the six places.
\backslashbox{From}{To}La Pedrera ( \(L\) )Nou Camp (N)Olympic Village ( \(O\) )Park Guell (P)Ramblas (R)Sagrada Familia ( \(S\) )
La Pedrera \(( L )\)-3530303735
Nou Camp \(( N )\)25-20212540
Olympic Village ( \(O\) )1540-253029
Park Guell ( \(P\) )303525-3520
Ramblas ( \(R\) )20301725-25
Sagrada Familia ( \(S\) )2535292030-
  1. Find the total travelling time for:
    1. the route \(L N O L\);
    2. the route \(L O N L\).
  2. Give an example of a Hamiltonian cycle in the context of the above situation.
  3. Salvadore intends to travel from one place to another until he has visited all of the places before returning to his starting place.
    1. Show that, using the nearest neighbour algorithm starting from Sagrada Familia \(( S )\), the total travelling time for Salvadore is 145 minutes.
    2. Explain why your answer to part (c)(i) is an upper bound for the minimum travelling time for Salvadore.
    3. Salvadore starts from Sagrada Familia ( \(S\) ) and then visits Ramblas ( \(R\) ). Given that he visits Nou Camp \(( N )\) before Park Guell \(( P )\), find an improved upper bound for the total travelling time for Salvadore.
AQA D1 2006 January Q9
9 A factory makes three different types of widget: plain, bland and ordinary. Each widget is made using three different machines: \(A , B\) and \(C\). Each plain widget needs 5 minutes on machine \(A , 12\) minutes on machine \(B\) and 24 minutes on machine \(C\). Each bland widget needs 4 minutes on machine \(A , 8\) minutes on machine \(B\) and 12 minutes on machine \(C\). Each ordinary widget needs 3 minutes on machine \(A\), 10 minutes on machine \(B\) and 18 minutes on machine \(C\). Machine \(A\) is available for 3 hours a day, machine \(B\) for 4 hours a day and machine \(C\) for 9 hours a day. The factory must make:
more plain widgets than bland widgets;
more bland widgets than ordinary widgets.
At least \(40 \%\) of the total production must be plain widgets.
Each day, the factory makes \(x\) plain, \(y\) bland and \(z\) ordinary widgets.
Formulate the above situation as 6 inequalities, in addition to \(x \geqslant 0 , y \geqslant 0\) and \(z \geqslant 0\), writing your answers with simplified integer coefficients.
(8 marks)
AQA D1 2007 January Q1
1 The following network shows the lengths, in miles, of roads connecting nine villages.
\includegraphics[max width=\textwidth, alt={}, center]{e47eb41e-0a4b-4865-a8ff-6c9978495ee0-02_856_1251_568_374}
  1. Use Prim's algorithm, starting from \(A\), to find a minimum spanning tree for the network.
  2. Find the length of your minimum spanning tree.
  3. Draw your minimum spanning tree.
  4. State the number of other spanning trees that are of the same length as your answer in part (a).
AQA D1 2007 January Q2
2 Five people \(A , B , C , D\) and \(E\) are to be matched to five tasks \(R , S , T , U\) and \(V\).
The table shows the tasks that each person is able to undertake.
PersonTasks
\(A\)\(R , V\)
\(B\)\(R , T\)
\(C\)\(T , V\)
\(D\)\(U , V\)
\(E\)\(S , U\)
  1. Show this information on a bipartite graph.
  2. Initially, \(A\) is matched to task \(V , B\) to task \(R , C\) to task \(T\), and \(E\) to task \(U\). Demonstrate, by using an alternating path from this initial matching, how each person can be matched to a task.
AQA D1 2007 January Q3
3 Mark is driving around the one-way system in Leicester. The following table shows the times, in minutes, for Mark to drive between four places: \(A , B , C\) and \(D\). Mark decides to start from \(A\), drive to the other three places and then return to \(A\). Mark wants to keep his driving time to a minimum.
\backslashbox{From}{To}\(\boldsymbol { A }\)\(\boldsymbol { B }\)\(\boldsymbol { C }\)\(\boldsymbol { D }\)
A-8611
B14-1325
C149-17
\(\boldsymbol { D }\)261018-
  1. Find the length of the tour \(A B C D A\).
  2. Find the length of the tour \(A D C B A\).
  3. Find the length of the tour using the nearest neighbour algorithm starting from \(A\).
  4. Write down which of your answers to parts (a), (b) and (c) gives the best upper bound for Mark's driving time.
AQA D1 2007 January Q4
4
  1. A student is using a bubble sort to rearrange seven numbers into ascending order.
    Her correct solution is as follows:
    Initial list18171326101424
    After 1st pass17131810142426
    After 2nd pass13171014182426
    After 3rd pass13101417182426
    After 4th pass10131417182426
    After 5th pass10131417182426
    Write down the number of comparisons and swaps on each of the five passes.
  2. Find the maximum number of comparisons and the maximum number of swaps that might be needed in a bubble sort to rearrange seven numbers into ascending order.
AQA D1 2007 January Q5
5 A student is using the following algorithm with different values of \(A\) and \(B\).
Line 10Input \(A , B\)
Line 20Let \(C = 0\) and let \(D = 0\)
Line 30Let \(C = C + A\)
Line 40Let \(D = D + B\)
Line 50If \(C = D\) then go to Line 110
Line 60If \(C > D\) then go to Line 90
Line 70Let \(C = C + A\)
Line 80Go to Line 50
Line 90Let \(D = D + B\)
Line 100Go to Line 50
Line 110Print \(C\)
Line 120End
    1. Trace the algorithm in the case where \(A = 2\) and \(B = 3\).
    2. Trace the algorithm in the case where \(A = 6\) and \(B = 8\).
  2. State the purpose of the algorithm.
  3. Write down the final value of \(C\) in the case where \(A = 200\) and \(B = 300\).
AQA D1 2007 January Q6
6 [Figure 1, printed on the insert, is provided for use in this question.]
Dino is to have a rectangular swimming pool at his villa.
He wants its width to be at least 2 metres and its length to be at least 5 metres.
He wants its length to be at least twice its width.
He wants its length to be no more than three times its width.
Each metre of the width of the pool costs \(\pounds 1000\) and each metre of the length of the pool costs \(\pounds 500\). He has \(\pounds 9000\) available. Let the width of the pool be \(x\) metres and the length of the pool be \(y\) metres.
  1. Show that one of the constraints leads to the inequality $$2 x + y \leqslant 18$$
  2. Find four further inequalities.
  3. On Figure 1, draw a suitable diagram to show the feasible region.
  4. Use your diagram to find the maximum width of the pool. State the corresponding length of the pool.
AQA D1 2007 January Q7
7 [Figure 2, printed on the insert, is provided for use in this question.]
The network shows the times, in seconds, taken by Craig to walk along walkways connecting ten hotels in Las Vegas.
\includegraphics[max width=\textwidth, alt={}, center]{e47eb41e-0a4b-4865-a8ff-6c9978495ee0-07_1435_1267_525_351} The total of all the times in the diagram is 2280 seconds.
    1. Craig is staying at the Circus ( \(C\) ) and has to visit the Oriental ( \(O\) ). Use Dijkstra's algorithm on Figure 2 to find the minimum time to walk from \(C\) to \(O\).
    2. Write down the corresponding route.
    1. Find, by inspection, the shortest time to walk from \(A\) to \(M\).
    2. Craig intends to walk along all the walkways. Find the minimum time for Craig to walk along every walkway and return to his starting point.
AQA D1 2007 January Q8
8
  1. The diagram shows a graph \(\mathbf { G }\) with 9 vertices and 9 edges.
    \includegraphics[max width=\textwidth, alt={}, center]{e47eb41e-0a4b-4865-a8ff-6c9978495ee0-08_188_204_411_708}
    \includegraphics[max width=\textwidth, alt={}, center]{e47eb41e-0a4b-4865-a8ff-6c9978495ee0-08_184_204_415_1105}
    \includegraphics[max width=\textwidth, alt={}, center]{e47eb41e-0a4b-4865-a8ff-6c9978495ee0-08_183_204_612_909}
    1. State the minimum number of edges that need to be added to \(\mathbf { G }\) to make a connected graph. Draw an example of such a graph.
    2. State the minimum number of edges that need to be added to \(\mathbf { G }\) to make the graph Hamiltonian. Draw an example of such a graph.
    3. State the minimum number of edges that need to be added to \(\mathbf { G }\) to make the graph Eulerian. Draw an example of such a graph.
  2. A complete graph has \(n\) vertices and is Eulerian.
    1. State the condition that \(n\) must satisfy.
    2. In addition, the number of edges in a Hamiltonian cycle for the graph is the same as the number of edges in an Eulerian trail. State the value of \(n\).
AQA D1 2008 January Q1
1 Five people, \(A , B , C , D\) and \(E\), are to be matched to five tasks, \(J , K , L , M\) and \(N\). The table shows the tasks that each person is able to undertake.
PersonTask
\(A\)\(J , N\)
\(B\)\(J , L\)
\(C\)\(L , N\)
\(D\)\(M , N\)
\(E\)\(K , M\)
  1. Show this information on a bipartite graph.
  2. Initially, \(A\) is matched to task \(N , B\) to task \(J , C\) to task \(L\), and \(E\) to task \(M\). Complete the alternating path \(D - M \ldots\), from this initial matching, to demonstrate how each person can be matched to a task.
    (3 marks)
AQA D1 2008 January Q2
2 [Figure 1, printed on the insert, is provided for use in this question.]
The feasible region of a linear programming problem is represented by $$\begin{aligned} x + y & \leqslant 30
2 x + y & \leqslant 40
y & \geqslant 5
x & \geqslant 4
y & \geqslant \frac { 1 } { 2 } x \end{aligned}$$
  1. On Figure 1, draw a suitable diagram to represent these inequalities and indicate the feasible region.
  2. Use your diagram to find the maximum value of \(F\), on the feasible region, in the case where:
    1. \(F = 3 x + y\);
    2. \(F = x + 3 y\).
AQA D1 2008 January Q3
3 The diagram shows 10 bus stops, \(A , B , C , \ldots , J\), in Geneva. The number on each edge represents the distance, in kilometres, between adjacent bus stops.
\includegraphics[max width=\textwidth, alt={}, center]{92175666-ef7a-4dca-9cdb-ebde1b40b2c9-03_595_1362_422_331} The city council is to connect these bus stops to a computer system which will display waiting times for buses at each of the 10 stops. Cabling is to be laid between some of the bus stops.
  1. Use Kruskal's algorithm, showing the order in which you select the edges, to find a minimum spanning tree for the 10 bus stops.
  2. State the minimum length of cabling needed.
  3. Draw your minimum spanning tree.
  4. If Prim's algorithm, starting from \(A\), had been used to find the minimum spanning tree, state which edge would have been the final edge to complete the minimum spanning tree.
AQA D1 2008 January Q4
4 [Figure 2, printed on the insert, is provided for use in this question.]
The network shows 11 towns. The times, in minutes, to travel between pairs of towns are indicated on the edges.
\includegraphics[max width=\textwidth, alt={}, center]{92175666-ef7a-4dca-9cdb-ebde1b40b2c9-04_1762_1056_516_484} The total of all of the times is 308 minutes.
    1. Use Dijkstra's algorithm on Figure 2 to find the minimum time to travel from \(A\) to \(K\).
    2. State the corresponding route.
  2. Find the length of an optimum Chinese postman route around the network, starting and finishing at \(A\). (The minimum time to travel from \(D\) to \(H\) is 40 minutes.)
AQA D1 2008 January Q5
5 [Figure 3, printed on the insert, is provided for use in this question.]
  1. James is solving a travelling salesperson problem.
    1. He finds the following upper bounds: \(43,40,43,41,55,43,43\). Write down the best upper bound.
    2. James finds the following lower bounds: 33, 40, 33, 38, 33, 38, 38 . Write down the best lower bound.
  2. Karen is solving a different travelling salesperson problem and finds an upper bound of 55 and a lower bound of 45 . Write down an interpretation of these results.
  3. The diagram below shows roads connecting 4 towns, \(A , B , C\) and \(D\). The numbers on the edges represent the lengths of the roads, in kilometres, between adjacent towns.
    \includegraphics[max width=\textwidth, alt={}, center]{92175666-ef7a-4dca-9cdb-ebde1b40b2c9-05_451_1034_1160_504} Xiong lives at town \(A\) and is to visit each of the other three towns before returning to town \(A\). She wishes to find a route that will minimise her travelling distance.
    1. Complete Figure 3, on the insert, to show the shortest distances, in kilometres, between all pairs of towns.
    2. Use the nearest neighbour algorithm on Figure 3 to find an upper bound for the minimum length of a tour of this network that starts and finishes at \(A\).
    3. Hence find the actual route that Xiong would take in order to achieve a tour of the same length as that found in part (c)(ii).
AQA D1 2008 January Q6
6 A student is solving cubic equations that have three different positive integer solutions.
The algorithm that the student is using is as follows:
Line 10 Input \(A , B , C , D\)
Line \(20 \quad\) Let \(K = 1\)
Line \(30 \quad\) Let \(N = 0\)
Line \(40 \quad\) Let \(X = K\)
Line 50 Let \(Y = A X ^ { 3 } + B X ^ { 2 } + C X + D\)
Line 60 If \(Y \neq 0\) then go to Line 100
Line \(70 \quad\) Print \(X\), "is a solution"
Line \(80 \quad\) Let \(N = N + 1\)
Line 90 If \(N = 3\) then go to Line 120
Line \(100 \quad\) Let \(K = K + 1\)
Line 110 Go to Line 40
Line 120 End
  1. Trace the algorithm in the case where the input values are:
    1. \(A = 1 , B = - 6 , C = 11\) and \(D = - 6\);
    2. \(A = 1 , B = - 10 , C = 29\) and \(D = - 20\).
  2. Explain where and why this algorithm will fail if \(A = 0\).